When we deal with complex numbers in mathematics, we can observe complex roots occurring in conjugate pairs. That means there will be an opposite sign in the middle of binomial terms (roots). Conjugates in math are remarkably effective in rationalizing radical expressions and complex numbers. In this article, you will understand the meaning of conjugates, the formula to find the conjugate along with solved examples.
Conjugate Meaning
In maths, Conjugates are defined as a pair of binomials with identical terms but parting opposite arithmetic operators in the middle of these similar terms. A few more examples of pairs of conjugates are given below:
4 – 3i, 4 + 3i
p + q, p – q
√3 + 1, √3 – 1
We can also observe the conjugates in one of the algebraic identities, such as:
a^{2} – b^{2} = (a + b)(a – b)
Here, a + b and a – b are conjugate pairs.
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Conjugate of Complex Number
Suppose z = x + iy is a complex number, then the conjugate of z is denoted by \(\overline{z}\) and is written as \(\overline{z}=x-iy\)
A few examples are given below to understand the conjugate of complex numbers in a better way.
Complex number |
Conjugate complex number |
-2 + 9i |
-2 – 9i or -(2 + 9i) |
2 – i√5 |
2 + i√5 |
2 – ib |
2 + ib |
2√2 + i(3√3) |
2√2 – i(3√3) |
Some of the important properties of conjugate of a complex number are listed below:
Suppose \(z\) = \(a~+~ib\) be a complex number such that,
- \(z ~+~ \overline{z}\) = \(a ~+ ~ib~+ ~(a~ – ~ib)\) = \(2a\) which is a complex number having imaginary part as zero.
- \(Re(z ~+~ \overline{z})\) = \(2a\), \(Im(z ~+ ~\overline{z}) \) = \(0\)
- \(z ~-~ \overline{z}\) = \(a~ +~ ib~ – ~(a~ -~ ib)\) = \(2bi\)
- \(Re(z~ -~ \overline{z})\) = \(0\), \(Im(z~ -~ \overline{z})\) = \(2b\)
Conjugate of a Matrix
It is possible to find the conjugate for a given matrix by replacing each element of the matrix with its complex conjugate.
Mathematically, a conjugate matrix is a matrix \(\overline{A}\) obtained by replacing the complex conjugate of all the elements of the matrix A.
Let’s have a look at the example given below.
Example: Find the conjugate matrix of the matrix \(A=\begin{bmatrix} 1 – 5i & 0\\ 0 & 6 + 7i \end{bmatrix}\).
Solution:
Given,
\(A=\begin{bmatrix} 1 – 5i & 0\\ 0 & 6 + 7i \end{bmatrix}\)Now, the conjugates of each of the elements of matrix A are:
Conjugate of 0 is 0.
Conjugate of 1 – 5i is 1 + 5i.
Conjugate of 6 + 7i is 6 – 7i.
Therefore, the conjugate of A is \(\overline{A}=\begin{bmatrix} 1 + 5i & 0\\ 0 & 6 – 7i \end{bmatrix}\)
Conjugate Surds
A conjugate surd is a sum and the difference between two simple quadratic surds. Conjugate surds are also known as complementary surds.
Suppose 3√2 and √5 are two simple quadratic surds, then the conjugate surds can be written using the sum, and the difference of these surds as 3√2 + √5 and 3√2 – √5, respectively. Here, two surds (3√2 + √5) and (3√2 – √5) are conjugated.
Similarly, we can say that two surds (-3√7 + √5) and (-3√7 – √5) are conjugate.
The general form of two binomial quadratic surds (x√a + y√b) and (x√a – y√b) are called conjugate surds.
Frequently Asked Questions on Conjugates
What is a conjugate in maths?
In maths, Conjugates are defined as a pair of binomials with identical terms but parting opposite arithmetic operators in the middle of these similar terms. For example, p – q is the conjugate of p + q.
How do you find the conjugate?
The conjugate of a binomial can be found by changing the sign, i.e. from plus to minus or minus to plus in the middle of the two terms.
What is the conjugate of 1?
The conjugate of 1 is equal to 1, since 1 is considered as the real part of the complex number or a single term. Hence, change of sign is not applicable here.
What is the conjugate of Z?
Either ˉz or z∗ denotes the complex conjugate of z. The complex conjugate has the same real part as z and the imaginary part with the opposite sign. That means, if z = a + ib is a complex number, then z∗ = a − ib will be its conjugate. In the polar form of a complex number, the conjugate of re^iθ is given by re^−iθ.
What is the conjugate of -6 – 5i?
As we know, the conjugate of a binomial is equal to the binomial with the opposite sign in the middle of terms. Thus, the conjugate of -6 – 5i is -6 + 5i.